Slide 10.1 Chapter 10: Index numberswainwrig/5751/Chapter 10.pdf · Slide 10.3 Barrow, Statistics...

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Barrow, Statistics for Economics, Accounting and Business Studies, 5th edition © Pearson Education Limited 2009

Chapter 10: Index numbers

• Index numbers summarise a mass of information.

• For example, the Retail Price Index summarises

the prices of thousands of goods and shows the

trend over time.

• The index of industrial production summarises

the output of industry over time.

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A simple index number (the price of coal)

Year Price Index

2002 36.97 100.0 (= 36.97/36.97 × 100)

2003 34.03 92.0 (= 34.03/36.97 × 100)

2004 37.88 102.5 (= 37.88/36.97 × 100)

2005 44.57 120.6 Etc.

2006 43.63 118.0

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Re-basing the index to 2004

• The index is set to 2004=100

• The information is the same, presentation slightly different

Year Price Index

2002 36.97 97.6 (= 36.97/37.88 × 100)

2003 34.03 89.8 (= 34.03/37.88 × 100)

2004 37.88 100.0 (= 37.88/37.88 × 100)

2005 44.57 117.7 Etc.

2006 43.63 115.2

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A price index for energy

• What is the price of energy?

Year

Coal

(£/tonne)

Petroleum

(£/tonne)

Electricity

(£/MWh)

Gas

(£/therm)

2002 36.97 132.24 29.83 0.780

2003 34.03 152.53 28.68 0.809

2004 37.88 153.71 31.26 0.961

2005 44.57 204.28 42.37 1.387

2006 43.63 260.47 55.07 1.804

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The Laspeyres index

• Construct a ‘shopping basket’ of fuels:

Quantities of fuel used by industry, 2002

Coal (m. tonnes) 1.81

Petroleum (m. tonnes) 5.70

Electricity (m. MWh) 112.65

Gas (m. therms) 5641

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Cost of the shopping basket IN 2002

Total cost: £8,581.013 million

Price Quantity Price x quantity

Coal (£/tonne) 36.97 1.81 66.916

Petroleum (£/tonne) 132.24 5.70 753.768

Electricity (£/MWh) 29.83 112.65 3360.350

Gas (£/m. therms) 0.780 5641 4399.980

Total 8581.013

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Cost of the same basket in different years

Formula Cost

2002 p0q0 8581.01

2003 p1q0 8725.39

2004 p2q0 9887.15

2005 p3q0 13842.12

2006 p4q0 17943.65

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Construction of the Laspeyres index

Year Formula Index

2002 100

00

00

qp

qp

100 ( 8725.39 / 8581.01 100)

2003 100

00

01

qp

qp

101.68 ( 9887.15 / 8581.01 100)

2004 100

00

02

qp

qp

115.22 etc.

2005 100

00

03

qp

qp

161.31

2006 100

00

04

qp

qp

209.11

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The Laspeyres index

• The index for year n, using year 0 as base year is

• Problem: the shopping basket is fixed over time

10000

0

qp

qpP

nn

L

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The Paasche index

• Updates the basket each year

Coal (m.

tonnes)

Petroleum (m.

tonnes)

Electricity (m.

MWh)

Gas (m.

therms)

2003 1.86 6.27 113.36 5677

2004 1.85 6.45 115.84 5258

2005 1.79 6.57 118.52 5226

2006 1.71 6.55 116.31 4910

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The Paasche formula

• The Paasche index for year n compares the cost

of the year n basket at year n prices to its cost at

base year prices

1000

n

nnn

Pqp

qpP

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Calculation of the Paasche index

Cost of basket at

current prices: Cost at 2002 prices Index

2002 8581.01 8581.01 100

2003 8863.52 8707.50 101.79

2004 9735.60 8478.09 114.83

2005 13692.05 8546.72 160.20

2006 17043.52 8228.72 207.12

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Comparison of indexes

Laspeyres

index Paasche

index

2002 100 100 2003 101.68 101.79 2004 115.22 114.83 2005 161.31 160.20 2006 209.11 207.12

The Paasche index (usually) rises more slowly

because firms switch to cheaper fuels

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Quantity indexes

• Laspeyres index

• Paasche index

10000

0

pq

pqQ

nn

L

1000

n

nnn

Lpq

pqQ

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Calculation of Laspeyres index

pqn Index

2002 8581.01 100

2003 8707.50 101.47 (= 8707.5/8581.01 100)

2004 8478.09 98.80 (= 8478.09/8581.01 100)

2005 8546.72 99.60

2006 8228.72 95.89

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Calculation of the Paasche index

pn qn pn q0 Index

2002 8581.01 8581.01 100

2003 8863.52 8725.39 101.58

2004 9735.60 9887.15 98.47

2005 13692.05 13842.12 98.92

2006 17043.52 17943.65 94.98

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An expenditure index

• Compares expenditure in different years

• Compares the cost of the year n basket at year n

prices to the base year basket at base year

prices

Ep q

p q

n n n

0 0

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Calculation of the expenditure index

pn qn Index

2002 8581.01 100

2003 8863.52 103.29

2004 9735.60 113.46

2005 13692.05 159.56

2006 17043.52 198.62

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Relationships between indexes

• In general, price quantity = expenditure

• So with index numbers, but note:

– Laspeyres price index Paasche quantity index =

expenditure index

– Paasche price index Laspeyres quantity index =

expenditure index

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Deflating a series

hence

• Dividing an expenditure series by a price index

gives a quantity index

n

P

n

L

n QPE

n

L

nn

PP

EQ

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Deflating energy expenditure

Expenditure at

current prices

Laspeyres price

index

Expenditure in

volume terms Index

2002 8581.01 100 8581.01 100

2003 8863.52 101.68 8716.86 101.58

2004 9735.60 115.22 8449.49 98.47

2005 13692.05 161.31 8487.99 98.92

2006 17043.52 209.11 8150.55 94.98

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Combining indexes - the chain index

‘Old’ index ‘New’ index Chain index

1998 104.54 104.54

1999 100 100

2000 104.63 104.63

2001 116.68 116.68

2002 111.87 100 111.87

2003 101.68 113.75

2004 115.22 128.90

2005 161.31 180.46

2006 209.11 233.93

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Summary

• Index numbers usefully summarise information

• The Laspeyres and Paasche methods are commonly

used to construct index numbers

• A price index can be used for deflating a series of

expenditures to obtain a quantity index (or ‘real’

expenditures)

Slide 10.1 Chapter 10: Index numberswainwrig/5751/Chapter 10.pdf · Slide 10.3 Barrow, Statistics...

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